Heat Transmission in Radiant Sections of Tube Stills D. W. WILSONAND W. E. LOBO,hl. W. Kellogg Co., New York, N. Y.,
AND
H. C. HOTTEL
Massachusetts Institute of Technology, Cambridge, Mass.
T
HE need of the oil inthat a temperature may be asA reciew qf empirical formulas for predicting signed to the flame equal to a dustry for an accurate heat transfer i n the combustion chambers of steamand, if possible, a quick mean of theoretical flame temboiler furnaces is followed by a description of i n e t h o d of p r e d i c t i n g heat perature and exit-gas temperatwelve tube-still furnaces widely different in type, transmission in the radiant secture), the problem is still comon which sixty-two performance tests are availtions of tube stills is obvious. p l i c a t e d . Such a method of s o l u t i o n h a s b e e n advanced It is the purpose of this paper able. Operating conditions include furnaces with by a number of investigators to discuss t h e m e c h a n i s m of and without air preheat, with and without flue-gas (3, I O ) , but it involves a triallieat transmission in c o m b u s recirculation, fired with gas or with oil, and with a n d - e r r o r s o l u t i o n of simultion chambers, to review briefly a wide range of variation of excess air. The data the empirical equations applitaneous equations, and is open are correlated on carious bases, the final recomcable in allied fields, a n d to to several objections from the standpoint of engineering design Iresent data on twelve different mended equation for box-type furnaces with tubes work. furnaces of widely varying types, in one row being It a p p e a r s , then, t h a t any correlated on various bases for suitably simple formula reprethe purpose of arriving a t a suit1 senting the effect of a series of able design equation. I.(= Gdo/Ac phenomena not in t h e m s e l v e s The evaluation of lieat transs i m p l e m u s t be to a large mission to the cold surfaces in a 3200 extent e m p i r i c a l . Since the combustion chamber is compliin which p equals the fraction of the available empirical method has been apcated by the inherently compliplied with considerable success to cated nature of the phenomena heat in the fuel, air, and recirculated flue gas the parallel problem of performr e s p o n s i b l e . The c h e m i c a l which is absorbed by the radiant tubes; G is ance of the combustion chamunion of molecules in the flame the air-fuel ratio; Q is the net heating value of bers of boiler furnaces, the results is attended by a certain amount the fuel $red in 1 hour; and A , is the total outof various investigators in that of heat radiation, of magnitude side tube area exposed to radiation. For a wider fieldmay be profitably considered dependent on the composition of as a guide in seeking a suihble the fuel, the maximum temperarange of furnace type (particularly for furnaces equation for tube-still furnace ture attained, and the absorbing with more than one row of tubes) a somewhat more design. There follow s e v e r a l cliaracteristics of the flame for its complicated equation is recommended involving a n such empirical treatments. own radiation. Superimposed “effective area” term instead of the actual tube on this, and of much greater DISCUSSION OF EhlPIRICAL area used in the preceding equation. magnitude, is the radiation due EQUATIOSS t o t h e h o t p r o d u c t s o f comHudson (7’)) considering data on several different types of I,ustion, particularly water vapor and carbon dioxide. This quantity IS dependent on the concentration of combustion steam-boiler furnaces, concluded that, of the total available products, the shape and size of the furnace chamber. and heat input to the furnace (including the net heat of combustion the mean temperature of the gases in the chamber. A third of the fuel plus the preheat in the air minus any loss in unsource of radiation from the flame is its luminosity or soot burned carbon monoxide, minus the heat content of the content, dependent in a complicated way on burner design, combustion products between steam temperature and room ~ ~ r i m a rand y secondary agration, and combustion chamber temperature) the fraction, pl, which is absorbed by the design. Finally, a considerable amount of heat is transmitted ultimate heat-receiving surfaces (in his case, water-cooled by direct convection, dependent on local temperature differ- tubes) is a simple function of two terms-the specific firing ence between gas and surface, turbulence of the gas, etc. By rate, C (expressed as pounds of fuel per square foot of waterall four of these mechanisms, heat is transferred from the flame cooled surface per hour), and the air-fuel ratio, G. The simple or gases to the various bounding surfaces of the system, equation which he proposed may be expressed in the following whether refractory or oil-cooled. Each element of refractory form: surface attains thereby an equilibrium temperature such that 1 P1 = its reception of heat from the gases by all mechanisms, less GC its external losses through the wall, is equal to its net radiation l+Gi through the gases to the various surfaces which it sees. This Orrok (9), with additional boiler furnace data available, latter radiation is dependent on angle factors between the refractory surface element and the surfaces surrounding it, proposed a modification of Hudson’s formula which fitted on the opacity of the gas to radiation from the surfaces, and the data much better. It is: on the emissivity of the surfaces. 1 IJ= Obviously any attempt to evaluate the net heat transferred GdFO in a combustion chamber by following in detail the above l + 27 actual mechanisms involved would be inordinately tedious if where U , = fractional heat transferred (not above temperature possible a t all. Even after a number of simplifying assumplevel of steam, but above atmospheric temperature) tions are made (such as that all refractory surfaces may be Co = specific firing rate, lb. of ‘le uivalent good bituminous coal”/sq. ft. of water-coied surface/hour assumed to attain the same equilibrium temperature and
‘’
486
I N D L S T R I A L A N D E N G I N E E R I N G C H E $1I S T li Y
May, 1932
The empirical nature of the Orrok modification of the Hudson equation becomes apparent when the effect of decreasing the firing rate, Co, is considered. the firing rate approaches zero, the products of combustion should have time to cool to the temperature of the water-cooled walls, and, external losses being neglected, the total heat of the fuel above water-cooled wall temperature should be transferred. According to Equation 2 , p becomes equal to unity, but the heat input is above inlet air temperature, not above tube-wall temperature. This factor, almost negligible in boiler furnaces, might be expected to be important in tube stills, in which the coolest surfaces present are at a temperature level of 800" F. or higher. Equation 1, therefore, is more rational with respect to the limit which p1 approaches than Equation 2, although the latter is somewhat simpler to evaluate. Artsay (1) later proposed an equation similar in some respects to that of Orrok, but slightly more complicated: 0.22Tg + T
1 PI =
no
(3)
35(.fqA)* "
I+-------
4%
where H , = useful heat input per lb. of roduct gas = (gross heating value of fuelr - (heat of unburned C, CO, HP,etc.) - (heat of vaporization of water) (sensible heat of air), all taken above a level equal to temp. of cooling medium in furnace, rind divided by number of pounds of products per lh. of fuel; B. t:u./ lb p, = fraction of above heat, H , , which is transmitted t(J
487
the heat available between absolute zero and T s o n the assumption of a constant specific heat uf 0.22. The term, p l , then, is the fraction (Jf the net heat input in fuel and air, a h \ e absolute zero, which is transmitted to the tIibe3. .Icc*ordingto Equation 3a, this fraction approachrs uiiil~v5the firing rate, ( I A , approaches zero, or RS the approaches infinity. It is obvious, then, that this instead of having a more rational basis of evaluati heat input than the Ormk-Hudson equation, ac less rational one. The rather strange definitioii of fuel factor, f, is & 6 ' 4 l , to comprehend. According to the definition, 1 m d e d propane (36 8 pounds) in 100 pounds of a gaseous would be counted as 1 mole of combustible in evaluati.1.g f p but 36 pounds of carbon and 8 pounds of hydrogen in e. ;alii or liquid fuel would be counted as 36/12 8/2, or 7 moles. CQ, more striking, the number of molecules in a unit weig1;t of fuel oil is to be considered nearly twice as great as the ni;:ilber in natural gas. Yo explanation is given of the basis for 1?1021 differentiation. iilthough ,4rtsay claims to have derive.? bh 3 general form of his equation from theoretical consideratiom,
+
+
drawing an analogy between heat content of gases
'1
1
+
tubes in combustion chamber
T s = temp. of cooling medium in tubes, ' P.ab:,. S, = projected surface exposed t o radiation, sq.ft. q.4
f
= =
weight of fuel burned in furnace, lb./sec. fuel factor: 1 for good coal, about 0.76 for natual gas, 1.5 for fuel oil; equals quantity (Ib. moles of combustiblein fuel per 100 lb. fue1)/8.67, defining moles as C, CO, Hp, and hydrocarbons
In this equation there is a return to Hudson's definitioii ul pl, and the objection to Orrok's basis of evaluating p (the
to evaluate the heat transmission in the former. procedure, although perhaps serving as an aid in vi the combustion chamber problem, in np sense cons proof of the validity of the equation. On the contraq , is excellent reawn t ( J expect that heat transmission in f chambers does not follow the laws of an impulse ti wholly on the babi- of its merit as a purdy empiricR1 tion. i t b similarity to that of Orrok. If 3600 (1-4 '8, (the pou~111,) fuel fired per hour per square foot of projected area) iq L . ~ ~ ' J ( ~ C'A, Equation 3a may be put in the form
fraction heat transmission) would seem to have been removed. The added term, 0.22TslH0, in the numerator, however, now keeps the combustion-chamber efficiency froin approaching its rational limit of unity as the firing rate, P A , approaches zero, or the surface, S,, approaches infinity. The equation may be converted to a form somewhat more readily interpreted. Multiplying both sides by Ho/(HO $. 0 . 2 2 T ~ ) , A variation in S, from 500 to 5000 square feet causes %1/10 to vary froin 1.86 to 2.34 (average 2.1). If C A varies from one obtains: 0.5 to 3.0, Ca1/lowill lie between 0.93 and 1.11 (average 1.09). If the fuel factor, f , is taken as unity, the above equation becomes By definition of terms, plHo is the heat transferred to the tubes per pound of gaseous products. The term, Ho 0.22Ts, is the heat, H o , initially available above the temperature, Ts, plus the term, 0.22T.9, which may be visualized as
+ The term, G (the air:fuel ratio of the Orrok equation), is replaced by 17, a fair figure for mean value of it. The numerical
TABLEI. CHARACTERISTICS OF FURNACES AND TESTS TUBE TUBESPACINQEXPOSED REFRACDIAM. CENTERTO TUBE TORY FURNACE OUTBIDE CENTER ARE.4 AREA Inches Inches Sq. It. s q . ft. 1 5.0 10.0 2390 1865 2 5.0 10.0 1889 1725 3 5.0 7.75 2456 2201 4 4.0 6.75 2279 1350
5 4.0 6 4.0 7 ... 8 5.0 9 5,O 10 4.0 11 5.0 12 4.0 T w o rows with center lines 2.62
7.12
8.75 1o:o 10.0
8.75 6.75
17.3"
inches apart.
564 556 1540 2951 1387 779 4443 1496
880 1627
1993 1565 1241 826 1676
VOL. OF
CHAMBER
cu. ft. 9920 7678 9741
5978 2050 4566 1500 16088 6737 3320 15100 7420
AIR PREHEAT
RECIRCULA-
Yes and no Yes Yes Yes
Yes and no Yes NO
No NO
Yes Yes and no Yes Yes Yes Y e8
TION
NO
NO No NO No Yes Yes No No
FUEL Gas or oil Gas Oil Gas Oil G as Gas GBB Gas Gas Gas Oil
I N D US T R I A L AND E S G I N E E R I N G CH E MISTR Y
488
TABLE11. NET HEATINQ
FURNaCE
TEST
FUEL
NET Hm.4~
VALUE LIBERAFUEL T I O N / H R . ( X lo-")" B. t . u . B. 1. u . OF
c.4LCULATIONS
FLUE GASA T
FROM
TEST DATA Gas
SEXBIBLE HEAT
TEMP.
BRIDQE WALL1
OVER
AIR/LB EXCESS BRIDQE FUELC AIRC WILL^ Lb. Lb. F. % 60.2 107.5 18.0 1140 9 44.9 49.0 30.4 1250 84 40.0 39.7 24.6 1270 49 38.7 35.5 21.1 1270 28 34.9 21.0 20.0 21 1305 34.4 22.0 17.5 1270 6 26.5 35.6 17.5 1275 6 35.6 31.0 20.6 1250 25 38.2 40.0 28.0 1250 70 40.4 48.0 30.5 1235 85 45.15 59.0 37.2 1225 125 68.05 66.0 17.7 1250 27 47.04 28.5 15.7 1300 13 42.7 26.5 18.0 1325 9 22.5 48.76 17.0 1400 3 47.5 21.0 17.0 1375 3 64 .? 68.5 17.3 1290 5 43.2 20.0 17.7 1345 7 45.8 33.0 25 1560 42.2 29.0 1580 79 32.5 37.0 1220 129 11.09 19.0 1355 29 21.97 38.5 .. 128 20.57 28.1 64 .. 3 9 23.83 25.8 50 .. 31.27 23.7 34 .. 35.75 26.7 52 .. r" 38.85 26.2 53 ._ 45.40 22.9 33 .. E 18.02 23.5 50 1543 E 46.93 27.4 1390 55 B 56.41 36.0 1430 105 50.32 24.1 1490 35 u 51.86 23.5 1490 33 bo 56.90 29.6 66 1430 64.96 24.8 1455 39 62.22 26.6 1440 50 0 60.56 27.6 1430 55 49 03 35.9 1400 119 52.50 31.5 1450 92 34.03 71.6 20.6 1241 35 16.00 137.0 1109 84 50.89 35.3 1466 101 54.73 35.8 1473 103 58.28 33.1 88 1477 24.58 31.6 123 1280 20.94 41.5 194 1250 r. 19 55 37.3 1160 165 1 23.55 22.5 1375 51 22.16 22.0 45 1380 25.46 24.3 1380 64 26.71 27.5 1405 88 U 27.78 27.9 1380 93 c 19.28 41.6 1200 194 20.23 40.2 1210 184 0 14.59 28.1 1170 91 M 14.30 25.4 1175 70 16.13 31.8 1180 120 17.84 33.6 1240 136 0 23.40 31.9 1240 120 k18.54 23.7 1230 59 1240 20.64 26.9 83 and recirculated flue gas, if any.
( X 10-8)
LB. F U E L b
Val. 24, No. 5
OILTEMP.
In
out
INTO
OIL/HR.
( x 10-96
HE~T INTO OIL CALCD.
FROM
FLUE
(GAS X 10-o)f SIDE
Lb./hr. F. F. B. t . u . B. 1. u . 1596 19.4 728 850 11.40 1825 19.4 700 15.40 865 1722 19.4 680 17.20 808 1700 19.4 675 875 18.30 19.4 1667 645 875 21.80 1643 19.4 648 872 20.30 1643 19.4 660 870 19.10 ... 1615 19.4 660 858 17.90 ... 1658 19.4 680 850 15.40 19.4 1676 685 850 15.00 1792 19.4 700 843 13.00 18.3 2405 718 805 16.60 2275 18.3 655 870 23.80 1970 19.4 650 858 22.90 ... 19.4 2305 650 26.40 885 ... 19.4 2387 655 885 25.60 . .. 2160 19.4 728 865 15.50 ... 2104 19.4 655 875 24.40 ... 1903 20.7 2 662 872 20.00 ... 2140 18.3 3 553 7 16 18.62 ... 1612 18.7 4 610 800 16.40 ,.. 605 18.3 5 286 5.64 ... 1100 19.5 6 598 818 4.76 ... 1040 19.5 510 788 7.10 ... 1206 19.5 494 778 8.59 ... 1530 19.5 480 789 10.90 ... 1750 19.5 497 786 11.64 ... 1970 19.5 512 786 12.71 ... 2290 19.5 473 775 15.23 ... 843 17.9 7 772 934 10.00 21.4 2040 8 749 927 20.04 24.97 2310 21.4 765 923 17.86 24.30 21.4 2200 758 947 25.62 27.74 21.4 2400 762 944 24.71 27.60 21.4 2610 764 944 24.45 26.74 21.4 ,3000 752 948 26.46 35.14 21.4 2860 749 948 26.85 31.71 21.4 2780 746 27.24 948 30.41 17.8 2150 742 15.69 878 19.62 17.8 2340 753 14.54 878 23.68 1225 17.9 9 729 850 8.35 365 17.9 10 702 810 2.80 ... 21.4 2150 11 774 1001 19.24 18.44 21.4 2290 787 1025 19.44 20.40 21.4 2440 1049 21.02 797 24.15 17.12 1240 1 12 800 580 8.81 11.14 2 17.12 1045 805 590 6.32 6.55 3 17.12 988 605 5.76 790 7.62 4 17.12 1238 585 810 12.25 10.78 5 17.12 1168 590 10.56 810 11.72 6 17.12 1312 10,12 600 810 12.36 17.12 1372 605 9.91 810 11.82 17.12 1418 610 9.66 810 12.09 92 1 17.12 605 800 9 5.08 7.35 17.12 10 973 605 810 5.37 7.73 17.12 11 570 810 752 6.22 7.55 17.12 12 6.48 560 728 810 7.91 17.12 813 13 585 810 7.75 5.83 17.12 590 882 14 6.01 820 8.01 17.12 690 1195 15 8.29 810 10.18 17.12 560 980 16 9.33 810 10.38 570 17.12 1057 810 8.98 17 10.36 a Includes net heat in fuel, preheated air, b Includes recirculated flue gas, if any. c At bridze wall: no differentiation as to source of air. d With hkh-velocity thermocouples. e Based on Fortsch and Whitman specific heat d a t a : on furnace 3, this item mcludes estimated heat of vaporization. J Allowance made for external losses; figures in this column are to be taken In preference t o those in preceding column when both are given, for reason8 discussed in text. 1
1 2 3 4 5 7 8 9 10 11 12 13 14 15 16 17 18 19 1 1 1 1 1 2 3 4 5 6 7 1 1 2 3 4 5 6 7 8 11 12 1 1 1 2 3
2r -
...
2
2
c
...
'G
;
4
similarity of the two equations is not unexpeckd, since Artsay obtained the constant 35 of Equation 3 by matching it to the Orrok equation in the mean range of data which the latter fitted. A third empirical treatment of heat transfer in the combustion chamber is that of DeBaufre ( 2 ) , which for the present purpose is best represented in the form, (4)
where q A,
heat transferred, B. t. u./hour tube area exposed t o radiation, sq. ft. T G = abs. temp. of prooducts of combustion leaving furnace chamber, F. 460 T s = abs. temp. of cooling medium inside tubes or cooled surfaces = =
+
The term E is called by DeBaufre the "effectiveness factor" of the water-cooled (oil-cooled) surfaces. It is actually a sort
of emissivity factor of the flame, and would be expected to have a maximum value of 1723 (the Stefan-Boltzmann constant) but for the fact that the radiating temperature of the flame is higher than the exit-gas temperature, TQ,used in the equation. DeBaufre's equation would have merit only if the factor E were predictable with some precision as a function of various operating variables. He suggests that it is a function of the rate of heat liberation per unit of furnace volume, but the correlation on such a basis of the data he presents is extremely poor, as indicated by his plots. If E were determinable, it would still be necessary for design purposes to introduce a second equation (a heat balance) to use in connection with Equation 4. DATA With the preceding empirical methods in mind, we may proceed with the attempt to correlate certain data available on tube-still combustion chambers. It should be borne in PRESENT.4TION AND CORRELATION O F
I S D U S T R I A L A N D E S G I N E E R I N G C H E h? I S T R Y
Ma), 1932
a5
06
07
oa
09
D
FIRING R A T E ,
C
-
FIGURE 1. PLOTOF
(l/p
20
3D
- l ) / G vs. C
mind that, in what follous, no pretense is made to completeness with respect to the methods of correlation tried. dlthough a n equation is developed which is believed to be better than anything a t present available for the radiant sections of tube stills, this presentation is to be considered a progress report on a project which is still being actively pursued by the authors. The data to be used are from twelve different furnaces of widely different types, operating under a wide range of conditions. Unfortunately all the data are not equally reliable; consequently it \vi11 be necessary to give considerably more weight to some tests than to others, Table I summarizes the characteristics of the different furnaces and tests, I t will be noted that there is a wide range of furnace t j p e from the standpoint both of exposed tube area and of ratio of refractory to exposed tube area. Operating conditions include tests with and nithout air preheat: with and without flue-gas recirculation; n i t h gas and n i t h 5.4 oil fuel; and n i t h a nide range of variation of the quantity-pounds of products per pound of fuel-accomplished by varying both recirculation and excess 50 air. Table I1 summarizes the calculations from test data and presents each test in sufficient detail to permit 4. its use by others interested in the problem of correlation. It is desirable, before proceeding with the analysis 42 of the data, to attempt to evaluate the reliability of the most important quantity obtained from the various 38 tests-the heat transfer to the tubes. I n general, the fuel quantity, oil flow, oil temperature, gas analyses, and gas temperatures are believed to be above the 13 standard of precision of most tests on tube stills-at least prohably sufficiently accurate to warrant the as- -13 sumption that major sources of error in final results lie 30 elsewhere. The desired heat transmission may be calculated either from the oil or from the gas side. Any 26 calculation based on oil-temperature change a ill involve, as questionable factors, the specific heat of oils at the high temperatures under c o n s i d e r a t i o n , the heat of 2 partial vaporization of the stock, and the heat of cracking. Any calculation based on the difference in heat 1. content of the entering fuel-air mixture and that of the flue gas over the bridge wall, with a correction for external losses in the combustion chamber, will involve as I possible sources of error the difficulty of obtaining truly r e p r e s e n t a t i v e gas composition and temperature over the bridge wall (a particularly great difficulty when air leakage in the radiant section is high) and the error in estimating external heat losses from the
489
furnace. I n the tests on furnaces 1-6,9, and 10, the heat transmission was calculated from the oil side for various reasons. In furnace 1,a calculation from the gas side would have introduced a n error proportional to the error in the gas recirculation ratio, the calculation of which \\as based on a heat balance on the convection section. Although the results of the eighteen tests on furnace 1 seem to be mutually consistent, the actual heat transmission may be as much as 5 per cent greater than the value obtained from the oil-temperature rise owing to heat of cracking or vaporization. Furnace 2 was treated the same way for the same reasonI. e., doubt as to recirculation ratio-and may also be in error 5 per cent. I n furnace 3 allow4D SO ance was made for vaporization. Although furnace 5 was a topping still, the vaporization was low (10 per cent) and was assumed to occur entirely beyond the radiant section. Results on furnace 6 were based on the oil side because bridge-wall temperatures were not available. Results on furnaces 9 and 10 were based on the sensible heat change of the oil because the total sensible heat picked up by the oil in the whole furnace was greater than the heat gilen up by the gas, even though the soaker section showed eridence of absorbing heat of cracking not allowed for in the calculation of heat picked up by the oil. These two tests, consequently, are of very questionable value for the present purpose. I n the tests on furnaces 7,8,11, and 12 the heat transmission was calculated from the flue-gas side. I n furnace 7 the oil meter was under suspicion. The tests on furnace 8 were carefully made TI ith conditions ideal for a reliable calculation based on the gas side. Air leakage into the radiant section was extremely small, outside Jyall temperatures were measured.
G C FIGURE2. PLOTOF
I/p
vs.
Gt/C
490
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 24, No. 3
tnminous coal per square foot of exposed tube area2 per hour. Morris and Rubin extended the meaning of the term G t o permit their equation to cover the case of flue-gas recirculation, by defining G as the pounds of air plus recirculated flue gas, per pound of fuel. Inasmuch as both the power to which C enters and the constant in the denominator are in doubt, a general equation (of which the Hudson equation, the Orrok-Hudson equation, and the Kellogg radiant-heat equation which Morris and Rubin developed, are variations) map he investigated :
or in more useful form for study.
(1p - 1)
"
'Clo
FIGURE 3. PLOTOF
l/p
vs. GdCo
The heat input to the oil, calculated from the gas side, averaged about 20 per cent greater than the calculated sensible heat change in the oil, the difference being attributed to heat of cracking and vaporization. I n furnace 11 the same procedure yielded heat transfer rates 8 per cent greater figured from the gas side than from the oil side (sensible heat only). Tests on furnaces 8 and 11 are considered the most reliable of the twelve. Although the performance of furnace 12 was calculated from the flue-gas side, the results (apparently consistent among themselves) are open to some question because of the large air leakage into the radiant section, making the flue-gas temperature and samples possibly not representative of gas a t the bridge wall. From the preceding discussion it is seen that too close an interpretation of the deviation of individual tests from any proposed method of predicting performance is not permissible. A deviation of 10 per cent may, in most cases, be attributed to poor method of correlation; a deviation of 5 per cent may not. An early attempt a t correlation of about half of the data now being considered was made by Morris and Rubin (8) who recommended an equation similar in form to that of Orrok. Their recommendation was
in which p and G are the same as in the Orrok equation, but C is the actual firing rate in pounds of fuel per square foot of total exposed' tube area per hour, whereas the COof Orrok's equation is the equivalent firing rate in pounds of good bi1 T h a t is, circumference of tube, times exposed length, times number of tubes.
/G
=
Cnk
A plot on log paper of ( l / p - 1)/G vb. C should permit a choice of n and k . Figure 1 presents the sixty-two data points so plotted. The dotted line represents Equation 5;s the dashed line represents the Orrok form of equat i o n , t h o u g h t h e definition of C is different from Orrok's. A line parallel to Orrok's-i. e., corresponding to a value of n of '/-represents the data better than the line with a slope of l / 3 ; a t least, the data from furnace 6 are brought into line with the ot,hers by such choice. Inasmuch as Figure 1 does not permit a direct determination of the error in determining p from the recommended curves, the data are shown in Figure 2 with l/p plotted against G d C The equation of the line on the plot is 1 P =
(GJ-$ I + - G d C + ___ 26
(7)
8000
The position of the line was chosen to give more weight to tests on furnaces 1 and 8 than to furnace 12. Except for one test Doint from furnace 6, the test on furnace 4,and the three tests- on furnace 11, t h e l i n e m a y be considered to represent the data with fair precision. The failure of the data from furnace 11 to 2 The area basis for Orrok's equation cannot be determined with oertainty from the text of his paper, though a statement c o n c e r n i n g his m e t h o d of evaluating the area of a particular f u r n a c e m a y be fairly interpreted as implying t h a t his d e f i n i t i o n of exposed area is identical with t h a t of Morris and Rubin for furnaces with b u t one row of tubes on a ad. 8 Not all the d a t a a p p e a r i n g on Figure 1 were a v a i l a b l e when Equation 5 was recommended.
0
